Dimension quotients as boundary limits: the general case
We identify, functorially, the boundary limit of a simply defined presentation functor with the dimension quotient, both for groups and for Lie rings over integers.
math.GR↗
arXiv subjects
Roman Mikhailov
Publications and source records attributed to Roman Mikhailov.
Two results on asphericity
We construct an explicit finite chain of non-aspherical group presentation complexes $K(\mathcal X)\subset K(\mathcal Y)\subset K(\mathcal Z)$ in which $K(\mathcal Y)$ is Cockcroft, both inclusion-induced maps on $\pi_2$ are zero, and the pair $(K(\mathcal Y),K(\mathcal X))$ has the identity property. This answers a question of Hitchman. The construction is given directly from a short balanced presentation of the binary icosahedral group. As the second result, we construct an integral Lie ring presentation in which a subpresentation has a nontrivial identity among relations, whereas the presentation itself has none.
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The Parafree Conjecture for associative algebras
For every base field, we construct a finitely generated parafree augmented associative algebra with countably infinite-dimensional second homology. This disproves the analogue of the Parafree Conjecture for associative algebras and answers a question of Ivanov and Lopatkin. Our example is the monoid algebra of a finitely generated but not finitely presented submonoid of a free monoid.
math.RA↗
Dimension quotients as boundary limits
For a functor from the category of free presentations of a group to the category of all groups we define the boundary limit as an image of the natural map from limit to colimit. We show that the fourth dimension quotient of a group can be naturally described as the boundary limit of a simply-defined functor.
math.GR↗
Limits via relations
In this paper, we study operations on functors in the category of abelian groups simplar to the derivation in the sense of Dold-Puppe. They are defined as derived limits of a functor applied to the relation subgroup over a category of free presentations of the group. The integral homology of the Eilenberg-Maclane space $K(\mathbb Z,3)$ appears as a part of description of these operations applied to symmetric powers.
math.KT↗
The Topology of Poker
We examine the complexity of the ``Texas Hold'em'' variant of poker from a topological perspective. We show that there exists a natural simplicial complex governing the multi-way winning probabilities between various hands, and that this simplicial complex contains $4$-dimensional spheres as induced subcomplexes. We deduce that evaluating the strength of a pair of cards in Texas Hold'em is an intricate problem, and that even the notion of who is bluffing against whom is ill-defined in some situations.
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HR-length of a free group via polynomial functors
We prove that for a subring $R\subseteq \mathbb Q$ and a free group $F$ of rank at least $2$ the length of the Bousfield's $HR$-localization tower for $F$ is at least $ω+ω$. The key ingredient of the proof is the theory of polynomial functors over $\mathbb Q.$
math.GR↗
Homotopy patterns in group theory
This is a survey. The main subject of this survey is the homotopical or homological nature of certain structures which appear in classical problems about groups, Lie rings and group rings. It is well known that the (generalized) dimension subgroups have complicated combinatorial theories. In this paper we show that, in certain cases, the complexity of these theories is based on homotopy theory. The derived functors of non-additive functors, homotopy groups of spheres, group homology etc appear naturally in problems formulated in purely group-theoretical terms. The variety of structures appearing in the considered context is very rich. In order to illustrate it, we present this survey as a trip passing through examples having a similar nature.
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Homology of the pronilpotent completion and cotorsion groups
For a non-cyclic free group $F$, the second homology of its pronilpotent completion $H_2(\widehat F)$ is not a cotorsion group.
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Right exact group completion as a transfinite invariant of the homology equivalence
We consider a functor from the category of groups to itself $G\mapsto \mathbb Z_\infty G$ that we call right exact $\mathbb Z$-completion of a group. It is connected with the pronilpotent completion $\hat G$ by the short exact sequence $1\to {\varprojlim}^1\: M_n G \to \mathbb Z_\infty G \to \hat G \to 1,$ where $M_n G$ is $n$-th Baer invariant of $G.$ We prove that $\mathbb Z_\infty π_1(X)$ is an invariant of homological equivalence of a space $X$. Moreover, we prove an analogue of Stallings' theorem: if $G\to G'$ is a 2-connected group homomorphism, then $\mathbb Z_\infty G\cong \mathbb Z_\infty G'.$ We give examples of $3$-manifolds $X,Y$ such that $ \hat{π_1(X)}\cong \hat{π_1( Y)}$ but $\mathbb Z_\infty π_1(X)\not \cong \mathbb Z_\infty π_1(Y).$ We prove that for a finitely generated group $G$ we have $(\mathbb Z_\infty G)/ γ_ω= \hat G.$ So the difference between $\hat G$ and $\mathbb Z_\infty G$ lies in $γ_ω.$ This allows us to treat $\mathbb Z_\infty π_1(X)$ as a transfinite invariant of $X.$ The advantage of our approach is that it can be used not only for $3$-manifolds but for arbitrary spaces.
math.AT↗
An odd order group without the dimension property
We construct a finitely presented group $G$ such that the $7$th dimension quotient $G\cap(1+\varpi(\mathbb Z G)^7)/γ_7(G)$ has an element of order $3$; this immediately leads to a finite $3$-group without the dimension property. This contradicts a series of results by N. Gupta.
math.GR↗
Group and Lie algebra filtrations and homotopy groups of spheres
We establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the "dimension problem" by providing a converse to Sjogren's theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by embedding for arbitrary $s,d$ the torsion of the homotopy group $π_s(S^d)$ into a dimension quotient, via a result of Wu. In particular, this invalidates some long-standing results in the literature, since for every prime $p$, there is some $p$-torsion in $π_{2p}(S^2)$ by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group $π_4(S^2)=\mathbb Z/2\mathbb Z$. We finally obtain analogous results in the context of Lie rings: for every prime $p$ there exists a Lie ring with $p$-torsion in some dimension quotient.
math.GR↗
Homological properties of parafree Lie algebras
In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated Lie algebra over $\mathbb Z$ is greater than two. Moreover, it is proven that there exists a countable parafree group with nontrivial $H_2$.
math.RA↗
A finite Q-bad space
We prove that for a free noncyclic group $F$, $H_2(\hat F_\mathbb Q, \mathbb Q)$ is an uncountable $\mathbb Q$-vector space. Here $\hat F_\mathbb Q$ is the $\mathbb Q$-completion of $F$. This answers a problem of A.K. Bousfield for the case of rational coefficients. As a direct consequence of this result it follows that, a wedge of circles is $\mathbb Q$-bad in the sense of Bousfield-Kan. The same methods as used in the proof of the above results allow to show that, the homology $H_2(\hat F_\mathbb Z,\mathbb Z)$ is not divisible group, where $\hat F_\mathbb Z$ is the integral pronilpotent completion of $F$.
math.AT↗
Limits, standard complexes and fr-codes
For a strongly connected category $\mathcal C$ with pair-wise coproducts, we introduce a cosimplicial object, which serves as a sort of resolution for computing higher derived functors of ${\sf lim} : \mathrm{Ab}^{\mathcal C}\to \mathrm{Ab}$. Applications involve Künneth theorem for higher limits and ${\sf lim}$-finiteness of ${\bf fr}$-codes. A dictionary for the ${\bf fr}$-codes with words of length $\leq 3$ is given.
math.GR↗
Higher colimits, derived functors and homology
A theory of higher colimits over categories of free presentations is developed. It is shown that different homology functors such as Hoshcshild and cyclic homology of algebras over a field of characteristic zero, simplicial derived functors, and group homology can be obtained as higher colimits of simply defined functors. Connes' exact sequence linking Hochschild and cyclic homology was obtained using this approach as a corollary of a simple short exact sequence. As an application of the developed theory it is shown that the third reduced $K$-functor can be defined as the colimit of the second reduced $K$-functor applied to the fibre square of a free presentation of an algebra. A Hopf-type formula for odd dimensional cyclic homology of an algebra over a field of characteristic zero is also proved.
math.KT↗
On homotopy aspects and cablings of virtual pure braid groups
By exploring simplicial structure of pure virtual braid groups, we give new connections between the homotopy groups of the 3-sphere and the virtual braid groups that are related to the theory of Brunnian virtual braids. The group structure of VP_n with n > 4 is determined by VP_3, VP_4 and virtual cablings given by iterated degeneracy operations on the generators and defining relations. The complete proofs will be published in the two forthcoming papers.
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An example of a fractal finitely generated solvable group
A finitely generated solvable group with unbounded iterated identity is constructed.
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